Etudes des proprietes des neutrinos dans les contextes ...

tel-00450051, version 1 - 25 Jan 2010
Figure C.1: Flavour oscillation as a spin precession. Adapted from []
The evolution equation will then be:
∂tP = B × P . (C.29)
As a consequence the Hamiltonian H can consequently be seen as a magnetic
field B for the polarization vector P. As one can see on Fig.(C.1), the vacuum
oscillation can be interpreted as spin precession around the vector B. The flavour
content of the vector P being its projection on the z − −axis. When θ0 is zero,
then no oscillation occur and the projection Pz is constant.
C.5 Equivalences among the formalisms
Using the notation where να means that a neutrino in the flavour α was created,
one can sum up in equation the equivalence by:
i ∂
⎛ ⎞ ⎛ ⎞
ψe,να ψe,να
⎝ ψµ,να
⎠ = H ⎝ ψµ,να
⎠ ←→i
∂t
∂
⎛
⎞ ⎛
Uee Ueµ Ueτ
⎝ Uµe Uµµ Uµτ ⎠ = H ⎝
∂t
ψτ,να
where να = νe, νµ, ντ.
ψτ,να
Uτe Uτµ Uττ
Uee Ueµ Ueτ
Uµe Uµµ Uµτ
Uτe Uτµ Uττ
(C.30)
One can also see easily the equivalence between the density matrix and the Bloch
vector formalism. Knowing that a Hermitian 2 × 2 matrix A is represented as
A =
Tr(A) + A · σ
2
, (C.31)
where σ is the vector of Pauli matrices and A the polarization vector, the commutation
relations of the Pauli matrices imply that an equation of motion of the
form
i∂tA = [B, C] (C.32)
173
⎞
⎠ .